Simulating Risk Measures via Asymptotic Expansions for Relative Errors

Abstract

Risk measures, such as value-at-risk and expected shortfall, are widely used in finance. With the necessary sample size being computed using asymptotic expansions for relative errors, we propose a general framework to simulate these risk measures for a wide class of dependent data. The asymptotic expansions are new even for independent and identical data. An extensive numerical study is conducted to compare the proposed algorithm against existing algorithms, showing that the new algorithm is easy to implement, fast and accurate, even at the 0.001 quantile level. Applications to the estimation of intra-horizon risk and to a comparison of the relative errors of value-at-risk and expected shortfall are also given.

Keywords: relative errors, importance sampling, geometric $\alpha$-mixing, order statistics, estimation time.

JEL Classification: C10 C15 C44 C53 C63 D81 G18 G28